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June 2017 A Note on the Shuffle Variant of Jeśmanowicz' Conjecture
Zsolt RÁBAI
Tokyo J. Math. 40(1): 153-163 (June 2017). DOI: 10.3836/tjm/1502179220

Abstract

Let $(a,b,c)$ be a primitive Pythagorean triple. In 1956, Jeśmanowicz conjectured that the equation $a^{x}+b^{y}=c^{z}$ has the unique solution $(x,y,z)=(2,2,2)$ in positive integers. In 2010 Miyazaki proposed a similar problem. He conjectured that if $(a,b,c)$ is again a primitive Pythagorean triple with $b$ even, then the equation $c^{x}+b^{y}=a^{z}$ with $x$, $y$ and $z$ positive integers has the unique solution $(x,y,z)=(1,1,2)$ if $c=b+1$ and no solutions if $c>b+1$. He also proved that his conjecture is true if $c \equiv 1 \pmod{b}$. We extend Miyazaki's result to the case $c \equiv 1 \pmod{b/2^{\textrm{ord}_{2}(b)}}$.

Citation

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Zsolt RÁBAI. "A Note on the Shuffle Variant of Jeśmanowicz' Conjecture." Tokyo J. Math. 40 (1) 153 - 163, June 2017. https://doi.org/10.3836/tjm/1502179220

Information

Published: June 2017
First available in Project Euclid: 8 August 2017

zbMATH: 06787092
MathSciNet: MR3689983
Digital Object Identifier: 10.3836/tjm/1502179220

Subjects:
Primary: 11D61

Rights: Copyright © 2017 Publication Committee for the Tokyo Journal of Mathematics

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Vol.40 • No. 1 • June 2017
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